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A293430
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Persistently squarefree numbers for base-2 shifting: Numbers n such that all terms in finite set [n, floor(n/2), floor(n/4), floor(n/8), ..., 1] are squarefree.
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7
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1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 23, 26, 29, 30, 31, 42, 43, 46, 47, 53, 58, 59, 61, 62, 85, 86, 87, 93, 94, 95, 106, 107, 118, 119, 122, 123, 170, 173, 174, 186, 187, 190, 191, 213, 214, 215, 237, 238, 239, 246, 247, 341, 346, 347, 349, 373, 374, 381, 382, 383, 426, 427, 429, 430, 431, 474, 478, 479
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OFFSET
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1,2
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COMMENTS
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Question: Is this sequence infinite? (My guess: yes). This is equivalent to questions asked in A293230. See also comments at A293441 and A293517.
For any odd n that is present, 2n is also present.
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LINKS
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EXAMPLE
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For 479 we see that 479 is prime (thus squarefree, in A005117), [479/2] = 239 is also a prime, [239/2] = 119 = 7*17 (a squarefree composite), [119/2] = 59 (a prime), [59/2] = 29 (a prime), [29/2] = 14 = 2*7 (a squarefree composite), [14/2] = 7 (a prime), [7/2] = 3 (a prime), [3/2] = 1 (the end of halving process 1 is also squarefree), thus all the values obtained by repeated halving were squarefree and 479 is a member of this sequence. Here [ ] stands for taking floor.
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MATHEMATICA
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With[{s = Fold[Append[#1, MoebiusMu[#2] #1[[Floor[#2/2]]]] &, {1}, Range[2, 480]]}, Flatten@ Position[s, _?(# != 0 &)]] (* Michael De Vlieger, Oct 10 2017 *)
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PROG
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(PARI)
is_persistently_squarefree(n, base) = { while(n>1, if(!issquarefree(n), return(0)); n \= base); (1); };
isA293430(n) = is_persistently_squarefree(n, 2);
n=0; k=1; while(k <= 10000, n=n+1; if(isA293430(n), write("b293430.txt", k, " ", n); k=k+1)); \\ Antti Karttunen, Oct 11 2017
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CROSSREFS
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Marked terms in the binary tree illustration of A293230.
Positions of nonzero terms in A293233.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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